A lapping or polishing machine can be used to finish a broad range of materials, e.g., glasses, ceramics, plastics, and metals. The item being finished (lapped or polished) rides on top of the finishing surface (lap), in which the lap is substantially larger than the item being lapped or polished. The lapping or polishing process gradually removes parts of the item to cause the item to have a desired surface profile. The removal process takes place as the result of mechanical interaction or a combination of chemical and mechanical interaction between the item being finished, an abrasive, and the lap surface. This process is generally rate determined by an equation involving both the relative speed between the lap and the finished item and the pressure at the interface. Lapping machine laps can be, e.g., cast iron. Polishing machine laps can include, e.g., any of a family of compounds known as “pitch” or any of a family of synthetic materials such as urethanes.